Concentration inequalities for $s$-concave measure

arXiv:0807.0080v1[math.PR]1Jul2008Concentrationinequalitiesfors-concavemeasuresofdilationsofBorelsetsandapplicationsM.FradeliziJuly1,2008Universit´eParis-EstLaboratoired’AnalyseetdeMath´ematiquesAppliqu´ees(UMR8050)Universit´edeMarnelaVall´ee77454MarnelaVall´eeCedex2,[e-mail]nequalityconjecturedbyBobkovonthemeasureofdilationsofBorelsetsinRnbyas-concaveprobability.OurresultgivesacommongeneralizationofaninequalityofNazarov,SodinandVolbergandaconcentrationinequalityofGu´edon.Applyingourinequal-itytothelevelsetsoffunctionssatisfyingaRemeztypeinequality,wededuce,asitisclassical,thatthesefunctionsenjoydimensionfreedistri-butioninequalitiesandKahane-Khintchinetypeinequalitieswithpositiveandnegativeexponent,withrespecttoanarbitrarys-concaveprobability.Keywords:dilation,localizationlemma,Remeztypeinequalities,log-concavemea-sures,largedeviations,smalldeviations,Khintchinetypeinequalities,sublevelsetsAMS2000SubjectClassification:46B07;46B09;60B11;52A20;26D0511IntroductionThemainpurposeofthispaperistoestablishasharpinequality,conjecturedbyBobkovin[B3],comparingthemeasureofaBorelsetinRnwithas-concaveprobabilityandthemeasureofitsdilation.Amongthes-concaveprobabilitiesarethelog-concaveones(s=0)andthustheGaussianones,sothatitisexpectedthattheysatisfygoodconcentrationinequalitiesandlargeandsmalldeviationsinequalities.ThisisindeedthecaseandtheseinequalitiesaswellasKahane-Khintchinetypeinequalitieswithpositiveandnegativeexponentarededuced.ByusingalocalizationtheoremintheformgivenbyFradeliziandGu´edonin[FG],weexactlydetermineamongs-concaveprobabilitiesμonRnandamongBorelsetsFinRn,withfixedmeasureμ(F),whatisthesmallestmeasureofthet-dilationofF(witht1).Thisinfimumisreachedforaone-dimensionalmeasurewhichiss-affine(seethedefinitionbelow)andF=[−1,1].Inotherterms,itgivesauniformupperboundforthemeasureofthecomplementofthedilationofFintermsoft,sandμ(F).Theresultinginequalityappliesperfectlytosublevelsetsoffunctionssatis-fyingaRemezinequality,i.e.functionssuchthatthet-dilationofanyoftheirsublevelsetsiscontainedinanotheroftheirsublevelsetinauniformway(seesection2.3below).Themainexamplesofsuchfunctionsfaretheseminorms(f(x)=kxkK,whereKisacentrallysymmetricconvexsetinRn),therealpoly-nomialsinn-variables(f(x)=P(x)=P(x1,...,xn),withP∈R[X1,...,Xn])andmoregenerallytheseminormsofvectorvaluedpolynomialsinn-variables(f(x)=kPNj=1Pj(x)ejkK,withP1,...,PN∈R[X1,...,Xn]ande1,...,eN∈Rn).Otherexamplesaregiveninsection3.Forthesefunctionswegetanupperboundforthemeasuresoftheirsublevelsetsintermsofthemeasureofothersublevelsets.Thisenablestodeducethattheysatisfylargedeviationinequal-itiesandKahane-Khintchinetypeinequalitieswithpositiveexponent.Butthemainfeatureoftheinequalityobtainedisthatitmayalsobereadbackward.ThusitalsoimpliessmalldeviationinequalitiesandKahane-Khintchinetypeinequalitieswithnegativeexponent.Beforegoinginmoredetailedresultsandhistoricalremarks,letusfixthenotations.GivensubsetsA,BoftheEuclideanspaceRnandλ∈R,wesetA+B={x+y;x∈A,y∈B},λA={λx;x∈A}andAc={x∈Rn;x/∈A}.Foralls∈(−∞,1],wesaythatameasureμinRniss-concaveiftheinequalityμ(λA+(1−λ)B)≥[λμs(A)+(1−λ)μs(B)]1/sholdsforallcompactsubsetsA,B⊂Rnsuchthatμ(A)μ(B)0andallλ∈[0,1].Thelimitcaseisinterpretedbycontinuity,thustherighthandsideofthisinequalityisequaltoμλ(A)μ1−λ(B)fors=0.Noticethatans-concavemeasureist-concaveforallt≤s.Foraprobabilityμ,supp(μ)denotesitssupport.Forγ∈(−1,+∞],afunctionf:Rn→R+isγ-concaveiftheinequalityf(λx+(1−λ)y)≥[λfγ(x)+(1−λ)fγ(y)]1/γ2holdsforallxandysuchthatf(x)f(y)0andallλ∈[0,1],wherethelimitcasesγ=0andγ=+∞arealsointerpretedbycontinuity,forexamplethe+∞-concavefunctionsareconstant.Thelinkbetweenthes-concaveprobabilitiesandtheγ-concavefunctionsisdescribedintheworkofBorell[Bor2].Theorem[Bor2]LetμbeameasureinRn,letGbetheaffinehullofthesupportofμ,setd=dimGandmtheLebesguemeasureonG.Thenfors≤1/d,μiss-concaveifandonlyifdμ=ψdm,where0≤ψ∈L1loc(Rn,dm)andψisγ-concavewithγ=s/(1−sd)∈(−1/d,+∞].Accordingtothistheorem,wesaythatameasureμiss-affinewhenitsdensityψsatisfiesthatψγ(orlogψifs=γ=0)isaffineonitsconvexsupportwithγ=s/(1−sd).In[Bor1],Borellstartedthestudyofconcentrationpropertiesofs-concaveprobabilities.HenoticedthatforanycentrallysymmetricconvexsetKtheinclusionKc⊃2t+1(tK)c+t−1t+1Kholdstrue.Fromthedefinitionofs-concavityhededucedthatforeverys-concavemeasureμμ(Kc)≥2t+1μ(tK)c
s+t−1t+1μ(K)s1/s.(1)Fromthisveryeasybutnon-optimalconcentrationinequality,BorellshowedthatseminormssatisfylargedeviationinequalitiesandKahane-Khintchinetypeinequalitieswithpositiveexponent.Thesamemethodwaspushedforwardin1999byLatala[L]todeduceasmallballprobabilityforsymmetricconvexsetswhichallowedhimtogetaKahane-Khintchineinequalityuntilthegeometricmean.In1991,Bourgain[Bou]usedtheKnothemap[K]totransportsublevelsetsofpolynomials.Hededucedthat,withrespectto1/n-concavemeasureonRn(i.e.uniformmeasureonconvexbodies),therealpolynomialsinn-variablessatisfysomenon-optimaldistributionandKahane-Khintchinetypeinequalitieswithpositiveexponent.ThesamemethodwasusedbyBobkovin[B2]andrecentlyin[B3]tog